[FOM] less and more news?!

[Harvey Friedman]

The previous statements in news?! again were not strong enough for
reversals. I fix them here, and add more material.

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It looks like simplifications are coming in on posting #192. I need to see
how far I can get, but here is what is happening.

Relation means "binary relation". We identify r tuples from Ek with rk
tuples from E.  

PROPOSITION 1. Let k,p >= 1 and R be a strictly dominating order invariant
relation on [1,2^p]k. There exists A containedin [1,2^p]k, such that every
element of Ak x [1,2^p]k is order equivalent to an element of Ak x (A U.
R[A]), relative to 1,2,4,...,2^p, in which 2^2^2^8k - 1 does not appear.

Proposition 1 is provably equivalent to the consistency of Mahlo cardinals
of finite order over ACA.

As in posting #192, 2^2^2^8k - 1 is meant to be a silly but safe number.
Also, as in posting #192, we can use

A delta R[A]

instead of 

A U. R[A]. 

Also note that if we remove

2^2^2^8k - 1 does not appear

in these statements, then they become easily provable.

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We now move to one dimensonal sets, using the well known semilinear function
concept. 

Let T:Nk into N. We say that T is strictly dominating if and only if for all
x in Nk, T(x) > max(x).

PROPOSITION 2. Let k,p >= 1 and T:[1,2^p]k into N be a strictly dominating
semilinear function with coefficients from [0,k]. There exists A containedin
[1,p!] containing (8k)!!!, such that A U. T[Ak] meets the image of Ak by any
semilinear function with coefficients from {0,1!,...,(p-1)!}.

We say that T is 2-expansive if and only if for all x, T(x) >= 2 max(x).

PROPOSITION 3. Let k,p >= 1 and T:[1,2^p]k into N be a 2-expansive
semilinear function with coefficients from [-k,k]. There exists A
containedin [-p!,p!] containing (8k)!!!, such that A U. T[Ak] meets the
image of Ak by any semilinear function with coefficients from
{0,+-1!,+-2!,...,+-(p-1)!}.

(8k)!!! is another silly safe number.

These appear to be equivalent to the consistency of Mahlo cardinals of
finite order, over ACA.

Harvey Friedman